## On convex and $$*$$-concave multifunctions.(English)Zbl 1085.39024

A characterization of *-concave and convex multifunctions which are continuous with respect to the Hausdorff metric is given. Namely, {a continuous multifunction} $$F:[a,b] \to \text{ clb}(Y)$$ is *-concave (resp. convex) if and only if for every $$s,t \in [a,b], \; s<t$$ the inclusion $\frac{1}{t-s} \int_{s}^{t} F(x)dx \subset (\text{resp.} \supset) \frac{F(s) \overset{*}{+} F(t)}{2}$ holds.
The main tool in the proof is the characterization of real convex function which is based upon Hadamard inequality.

### MSC:

 39B62 Functional inequalities, including subadditivity, convexity, etc. 26E25 Set-valued functions
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